A New Method for Obtaining Velocity and Diffusivity from Time-Dependent Distributions of a Tracer via the Maximum Likelihood Estimator for the Advection–Diffusion Equation

Abstract

An inverse problem for the advection–diffusion equation is considered, and a method of maximum likelihood (ML) estimation is developed to derive velocity and diffusivity from time-dependent distributions of a tracer. Piterbarg and Rozovskii showed theoretically that the ML estimator for diffusivity is consistent ever in an asymptotic case of infinite number of observational spatial modes. In the present work, the ML estimator is studied based on numerical experiments with a tracer in a two-dimensional flow under the condition of a limited number of observations in space. The numerical experiments involve the direct and the inverse problems. For the former, the time evolution of a tracer is simulated using the Galerkin-type method—as a response of the conservation equation to stochastic forcing. In the inverse problem, the advection–diffusion equation is fitted to the simulated data employing the ML estimator. It is shown that the ML method allows us a method to estimate diffusion coefficient componentsDxandDybased on a short time series of tracer observations. The estimate of the diffusion anistropy,Dx/Dy, is shown to be even more robust than the estimate of the diffusivity itself. A comparison with an estimation technique based on the finite-difference approximation demonstrates advantages of the ML estimator. Finally, the ML method is employed for analysis of heat balance in the upper layer of the North Pacific in the winter. This application focuses on the heat diffusion anisotropy at the ocean mesoscale.